Problem: Solve for $x$ and $y$ by deriving an expression for $x$ from the second equation, and substituting it back into the first equation. $\begin{align*}4x-4y &= -6 \\ -5x+2y &= -2\end{align*}$
Answer: Begin by moving the $y$ -term in the second equation to the right side of the equation. $-5x = -2y-2$ Divide both sides by $-5$ to isolate $x$ $x = {\dfrac{2}{5}y + \dfrac{2}{5}}$ Substitute this expression for $x$ in the first equation. $4({\dfrac{2}{5}y + \dfrac{2}{5}}) - 4y = -6$ $\dfrac{8}{5}y + \dfrac{8}{5} - 4y = -6$ Simplify by combining terms, then solve for $y$ $-\dfrac{12}{5}y + \dfrac{8}{5} = -6$ $-\dfrac{12}{5}y = -\dfrac{38}{5}$ $y = \dfrac{19}{6}$ Substitute $\dfrac{19}{6}$ for $y$ in the top equation. $4x-4( \dfrac{19}{6}) = -6$ $4x-\dfrac{38}{3} = -6$ $4x = \dfrac{20}{3}$ $x = \dfrac{5}{3}$ The solution is $\enspace x = \dfrac{5}{3}, \enspace y = \dfrac{19}{6}$.